91Ó°ÊÓ

A continuous function \(f\) is defined on a closed, bounded interval \(I\). A consequence of the Intermediate Value Theorem is that there is an interval \([\alpha, \beta]\) with the property that, for every \(\gamma\) between \(\alpha\) and \(\beta,\) the equation \(f(x)=\gamma\) has a solution \(x=c\) with \(c\) in \(I .\) What are \(\alpha\) and \(\beta ?\) $$ f(x)=x+24 / x, \quad I=[2,6] $$

Express \(1.11111111 \ldots\) as a rational number.

In Exercises \(47-56,\) use your intuition to decide whether the limit exists. Justify your answer by using the rigorous definition of limit. $$ \begin{array}{l} \lim _{x \rightarrow-3} f(x) \text { where } \\ \qquad f(x)=\left\\{\begin{array}{cl} 6 & \text { if } x \leq-3 \\ -2 x & \text { if } x>-3 \end{array}\right. \end{array} $$

In Exercises \(67-73,\) use algebraic manipulation (as in Example 5 ) to evaluate the limit. $$ \lim _{h \rightarrow 0} \frac{h}{\sqrt{1+2 h}-1} $$

In Exercises \(74-80\), evaluate the one-sided limits. $$ \lim _{x \rightarrow 3^{-}} \frac{x-3}{|x-3|} $$

Graph the function \(f(x)=(1+x)^{1 / x}\) for \(0